# Exercici 10.18

## Apartat a)

kill(all)$

'Jz(t) = Jz:matrix([Dz(t)]);

u(t):=t*sin(t)$       diff(u(t), t);
v(t):=exp(2*t)$     diff(v(t), t);

Du = Du:sin(t)+t*cos(t);
Dv = Dv:2*%e^(2*t);

Jgt = Jgt: matrix ( [Du, Dv] );

'Jf(x,y) = Jf: matrix ( [Dxf] , [Dyf]);

Dz(t) = Jgt.Jf;

## Apartat b)

kill(all)$

'Ju(t) = Ju:matrix([Du(t)]);

x(t):=sin(2*t)$       
Dx=Dx: diff(x(t), t);
y(t):=exp(2*t^2)$     
Dy=Dy: diff(y(t), t);

Jgt = Jgt: matrix ( [Dx, Dy] );

f(x,y):= x^2 + 2*y^2;

Jf(x,y) = Jf: jacobian( [f(x,y)] , [x,y]);

Dxf(x,y):=2*x;      Dyf(x,y):=4*y;

Dxf(x(t), y(t));
Dyf(x(t), y(t));

'Du(t) = Dxf(x(t), y(t))*Dx + Dyf(x(t), y(t))*Dy;

## Apartat c)

kill(all)$

x=x:2*u*v;   
y=y:u/(3*v);

Dux=Dux:diff(x,u,1);   Dvx=Dvx:diff(x,v,1);
Duy=Duy:diff(y,u,1);   Dvy=Dvy:diff(y,v,1);

'Jgt1 = Jgt1: matrix ( [Dux, Duy] );

'Jgt2 = Jgt2: matrix ( [Dvx, Dvy] );

z(x,y):=f(x,y);

'Jf(x,y) = Jf: matrix ( [Dxf] , [Dyf]);

Duz= Jgt1.Jf;

Dvz= Jgt2.Jf;